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Integral formulas and Hodge decomposition in the theory of generalized partial-slice mo-nogenic functions

Manjie Hu, Chao Ding

Abstract

This paper explores generalized slice monogenic functions by introducing their operator symbols, representation formula, and integral formula. The study extends the Teodorescu transform to a broader class of theorems and inferences, providing new analytical tools for function theory in this setting. Additionally, the Hodge decomposition is established, providing a foundation for further research.

Integral formulas and Hodge decomposition in the theory of generalized partial-slice mo-nogenic functions

Abstract

This paper explores generalized slice monogenic functions by introducing their operator symbols, representation formula, and integral formula. The study extends the Teodorescu transform to a broader class of theorems and inferences, providing new analytical tools for function theory in this setting. Additionally, the Hodge decomposition is established, providing a foundation for further research.
Paper Structure (7 sections, 17 theorems, 109 equations)

This paper contains 7 sections, 17 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Clifford algebras
  4. Monogenic and slice monogenic functions
  5. Generalized partial-slice monogenic functions
  6. Norm estimates of Teodorescu transform
  7. Hodge decomposition on a Banach space

Key Result

Theorem 3.5

Xu2 Let $f\in \mathcal{G} \mathcal{S} \left( \Omega _D \right)$. Then it holds that, for every $\boldsymbol{x}=\boldsymbol{x}_p+r\underline{\omega}\in \Omega _D$ with $\underline{\omega}\in \mathbb{S}$, for all $\underline{\omega}_1\ne \underline{\omega}_2\in \mathbb{S}$. In particular, if $\underline{\omega}_1=-\underline{\omega}_2 =\underline{\eta}\in \mathbb{S}$, we have

Theorems & Definitions (33)

  • Definition 2.1: Monogenic functions
  • Definition 2.2: Slice monogenic functions
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.5: Representation Formula
  • Definition 3.6
  • Theorem 3.7: Cauchy-Pompeiu formula
  • Proposition 3.8
  • ...and 23 more